fredag 20 januari 2017

Shaky Basis of Quantum Mechanics

Schrödinger's equation! Where did we get that equation from? Nowhere. It is not possible to derive it from anything you know. It came out of the mind of Schrodinger.  (Richard P. Feynman)

In the final analysis, the quantum mechanical wave equation will be obtained by a postulate, whose justification is not that it has been deduced entirely from information already known experimentally (Eisberg and Resnick in Quantum Physics)

Schrödinger's equation as the basic mathematical model of quantum mechanics is obtained as follows:

Start with classical mechanics with a Hamiltonian of the following form for a system of $N$ interacting point particles of unit mass with positions $x_n(t)$ and momenta $p_n=\frac{dx_n}{dt}$ varying with time $t$ for $n=1,...N$:
  • $H(x_1,...,x_N)=\frac{1}{2}\sum_{n=1}^Np_n^2+V(x_1,....,x_N)$     
where $V$ is a potential depending on the particle positions $x_n$, with the corresponding equations of motion
  • $\frac{dp_n}{dt}=\frac{\partial V}{\partial x_n}$ for $n=1,...,N$.           (1)
Proceed by formally replacing momentum $p_n$ by the differential operator $-i\nabla_n$ where $\nabla_n$ is the gradient operator acting with respect to $x_n$ now viewed as the coordinates of three-dimensional space (and $i$ is the imaginary unit), to get the Hamiltonian 
  • $H(x_1,...,x_N)=-\frac{1}{2}\sum_{n=1}^N\Delta_n +V(x_1,...,x_N)$
supposed to be acting on a wave function $\psi (x_1,...,x_N)$ depending on $N$ 3d coordinates $x_1,...,x_N$, where $\Delta_n$ is the Laplacian with respect to coordinate $x_n$.  Then postulate Schrödinger's equation with a vague reference to (1) as a linear multi-d equation of the form:
  • $i\frac{\partial \psi}{\partial t}=H\psi$.         (2)
Schrödinger's equation thus results from inflating single points to full 3d spaces in a purely formal twist of classical mechanics by brutally changing the meaning of $x_n$ from point to full 3d space and then twisting (1) as well. The inflation gives a wave function which depends on $3N$ space coordinates and as such has no physicality and is way beyond computability.

The inflation corresponds to a shift from actual position, which may be of interest, to possible position (which can be anywhere), which has no interest. 

The inflation from point to full 3d space has become the trade mark of modern physics as expressed in Schrödinger's multi-d linear equation, with endless speculation without conclusion about the possible physics of the inflation and the meaning of (2). 

The formality and lack of physicality of the inflation of course should have sent Schrödinger's multi-d linear equation (2) to the waste-bin from start, but it didn't happen with the argument that even if the physics of the equation was beyond rationale, predictions from the equation always (yes, always!!) agree with observation. The lack of scientific logic was thus acknowledged from start, but it was taken for granted that anyway the equation describes physics very accurately. If a prediction from computation with Schrödinger's equation does not compare well with observation, there must be something wrong with the computation or comparison, never with the equation itself...

But solutions of Schrödinger's multi-d equation cannot be computed in any generality and thus claims of general validity has no real ground. It is simply a postulate/axiom and as such true by assumption as a tautology which can only be true.

The main attempts to give the inflation of classical mechanics into Schrödinger's multi-d linear equation a meaning, are:
  • Copenhagen Interpretation CI (probabilistic)
  • Many World Interpretation MWI (infinitely many parallel universa in certain contact) 
  • Pilot-Wave (Bohm) 
with no one explanation gathering clear acceptance.   In particular,  Schrödinger did not like these interpretations of his equation and dreamed of a different version in 3d with physical "anschaulich" meaning, but did not find it...

In the CI the possibilities become an actualities by observation, while in MWI all possibilities are viewed as actualities and in Bohmian mechanics the pilot wave represents the possibilities with a particle somehow carried by the wave representing actuality...all very strange...        

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